What is the language of single cells? What are the elementary symbols of the code? Most typically, we think about the response as a firing rate, r(t), or a modulated spiking probability, P(r = spike|s(t)). Two extremes of description: A Poisson model, where spikes are generated randomly with rate r(t). However, most spike trains are not Poisson (refractoriness, internal dynamics). Fine temporal structure might be meaningful.  Consider spike patterns or “words”, e.g. symbols including multiple spikes and the interval between retinal ganglion cells: “when” and “how much”

Spike Triggered Average 2-Spike Triggered Average (10 ms separation) 2-Spike Triggered Average (5 ms) Multiple spike symbols from the fly motion sensitive neuron

Spike statistics Stochastic process that generates a sequence of events: point process Probability of an event at time t depends only on preceding event: renewal process All events are statistically independent: Poisson process Poisson: r(t) = r independent of time, probability to see a spike only depends on the time you watch. P T [n] = (rT) n exp(-rT)/n! Exercise: the mean of this distribution is rT the variance of this distribution is also rT. The Fano factor = variance/mean = 1 for Poisson processes. The Cv = coefficient of variation = STD/mean = 1 for Poisson Interspike interval distribution P(T) = r exp(-rT)

The Poisson model (homogeneous) Probability of n spikes in time T as function of (rate  T) Poisson approaches Gaussian for large rT (here = 10)

How good is the Poisson model? Fano Factor Fano factor Data fit to: variance = A  mean B A B Area MT

How good is the Poisson model? ISI analysis ISI Distribution from an area MT Neuron ISI distribution generated from a Poisson model with a Gaussian refractory period

How good is the Poisson Model? C V analysis Coefficients of Variation for a set of V1 and MT Neurons Poisson Poisson with ref. period

spike-triggering stimulus feature stimulus X(t) decision function spike output Y(t) x1x1 f1f1 P(spike|x 1 ) x1x1 Decompose the neural computation into a linear stage and a nonlinear stage. Modeling spike generation Given a stimulus, when will the system spike? To what feature in the stimulus is the system sensitive? Gerstner, spike response model; Aguera y Arcas et al. 2001, 2003; Keat et al., 2001 Simple example: the integrate-and-fire neuron

Let’s start with a rate response, r(t) and a stimulus, s(t). The optimal linear estimator is closest to satisfying Predicting the firing rate Want to solve for K. Multiply by s(t-  ’) and integrate over t: Note that we have produced terms which are simply correlation functions: Given a convolution, Fourier transform: Now we have a straightforward algebraic equation for K(w): Solving for K(t),

Predicting the firing rate For white noise, the correlation function C ss (  ) =    So K(  ) is simply C rs (  ). Going back to:

spike-triggering stimulus feature stimulus X(t) decision function spike output Y(t) x1x1 f1f1 P(spike|x 1 ) x1x1 The decision function is P(spike|x 1 ). Derive from data using Bayes’ theorem: P(spike|x 1 ) = P(spike) P(x 1 | spike) / P(x 1 ) P(x 1 ) is the prior : the distribution of all projections onto f 1 P(x 1 | spike) is the spike-conditional ensemble : the distribution of all projections onto f 1 given there has been a spike P(spike) is proportional to the mean firing rate Modeling spike generation

Models of neural function spike-triggering stimulus feature stimulus X(t) decision function spike output Y(t) x1x1 f1f1 P(spike|x 1 ) x1x1 Weaknesses

STA Gaussian prior stimulus distribution Spike-conditional distribution covariance Reverse correlation: a geometric view

Dimensionality reduction C ij = - - The covariance matrix is simply Properties: If the computation is low-dimensional, there will be a few eigenvalues significantly different from zero The number of eigenvalues is the relevant dimensionality The corresponding eigenvectors span the subspace of the relevant features Stimulus prior Bialek et al., 1997

Functional models of neural function spike-triggering stimulus feature stimulus X(t) decision function spike output Y(t) x1x1 f1f1 P(spike|x 1 ) x1x1

spike-triggering stimulus features stimulus X(t) multidimensional decision function spike output Y(t) x1x1 x2x2 x3x3 f1f1 f2f2 f3f3 Functional models of neural function

? spike-triggering stimulus features ? stimulus X(t) multidimensional decision function spike history feedback spike output Y(t) x1x1 x2x2 x3x3 f1f1 f2f2 f3f3 ? Functional models of neural function

Covariance analysis Let’s develop some intuition for how this works: the Keat model Keat, Reinagel, Reid and Meister, Predicting every spike. Neuron (2001) Spiking is controlled by a single filter Spikes happen generally on an upward threshold crossing of the filtered stimulus  expect 2 modes, the filter F(t) and its time derivative F’(t)

Covariance analysis

Let’s try some real neurons: rat somatosensory cortex (Ras Petersen, Mathew Diamond, SISSA: SfN 2003). Record from single units in barrel cortex

Covariance analysis Spike-triggered average: Pre-spike time (ms) Normalised velocity

Covariance analysis Is the neuron simply not very responsive to a white noise stimulus?

Covariance analysis PriorSpike- triggered Difference

Covariance analysis Pre-spike time (ms) Velocity EigenspectrumLeading modes

Input/output relations wrt first two filters, alone: and in quadrature: Covariance analysis

Pre-spike time (ms) Velocity (arbitrary units) How about the other modes? Next pair with +ve eigenvalues Pair with -ve eigenvalues Covariance analysis

Firing rate decreases with increasing projection: suppressive modes (Simoncelli et al.) Input/output relations for negative pair